Soliton cellular automata associated with infinite reduced words

TL;DR

This paper introduces a family of cellular automata linked to infinite reduced words in affine symmetric groups, analyzing their soliton solutions and revealing a duality with the ln-box-ball system.

Contribution

It defines new cellular automata associated with affine symmetric groups and explores their soliton solutions and duality with existing integrable systems.

Findings

Identification of soliton solutions for ln automata

Establishment of duality with ln-box-ball system

Connection to tropicalization of rational maps

Abstract

We consider a family of cellular automata $\Phi(n,k)$ associated with infinite reduced elements on the affine symmetric group $\hat S_n$, which is a tropicalization of the rational maps introduced by two of the authors. We study the soliton solutions for $\Phi(n,k)$ and explore a `duality' with the $\mathfrak{sl}_n$-box-ball system.